58 results
Dynamics and decay of a spherical region of turbulence in free space
- Ke Yu, Tim Colonius, D. I. Pullin, Grégoire Winckelmans
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- Journal:
- Journal of Fluid Mechanics / Volume 907 / 25 January 2021
- Published online by Cambridge University Press:
- 27 November 2020, A19
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We perform direct numerical simulation and large-eddy simulation of an initially spherical region of turbulence evolving in free space. The computations are performed with a lattice Green's function method, which allows the exact free-space boundary conditions to be imposed on a compact vortical region. Large-eddy simulations are conducted with the stretched vortex subgrid stress model. The initial condition is spherically windowed, isotropic homogeneous incompressible turbulence. We study the spectrum and statistics of the decaying turbulence and compare the results with decaying isotropic turbulence, including cases representing different low-wavenumber behaviour of the energy spectrum (i.e. $k^2$ versus $k^4$). At late times the turbulent sphere expands with both mean radius and integral scale showing similar timewise growth exponents. The low-wavenumber behaviour has little effect on the inertial scales, and we find that decay rates follow the predictions of Saffman (J. Fluid Mech., vol. 27, 1967, pp. 581–593) in both cases, at least until approximately $400$ initial eddy turnover times. The boundary of the spherical region develops intermittency and features ejections of vortex rings. These are shown to occur at the integral scale of the initial turbulence field and are hypothesized to occur due to a local imbalance of impulse on this scale.
Evolution of a shock generated by an impulsively accelerated, sinusoidal piston
- N. Shen, D. I. Pullin, R. Samtaney, V. Wheatley
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- Journal of Fluid Mechanics / Volume 907 / 25 January 2021
- Published online by Cambridge University Press:
- 26 November 2020, A35
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We consider the evolution of a shock wave generated by an impulsively accelerated, two-dimensional, almost planar piston with a sinusoidally corrugated surface of amplitude $\epsilon$. We develop a complex-variable formulation for a nonlinear theory of generalized geometrical shock dynamics (GGSD) (Best, Shock Waves, vol. 1, issue 4, 1991, pp. 251–273; Best, Proc. R. Soc. Lond. A, vol. 442, 1993, pp. 585–598) as a hierarchical expansion of the Euler equations that can be closed at any order. The zeroth-order truncation of GGSD is related to the equations of Whitham's geometrical shock dynamics (GSD), while higher-order corrections incorporate non-uniformity of the flow immediately behind the piston-driven shock. Numerical solutions to GGSD systems up to second order are coupled to an edge-detection algorithm in order to investigate the hypothesized development of a shock-shape curvature singularity as the rippled shock evolves. This singular behaviour, together with the simultaneous development of a Mach-number discontinuity, is found at all orders of the GGSD hierarchy for both weak and strong shocks. The critical time at which a curvature singularity occurs converges as the order of the GGSD system increases at fixed $\epsilon$, and follows a scaling inversely proportional to $\epsilon$ at sufficiently small values. This result agrees with the weakly nonlinear GSD analysis of Mostert et al. (J. Fluid Mech., vol. 846, 2018, pp. 536–562) for a general Mach-number perturbation on a planar shock, and suggests that this represents the universal behaviour of a slightly perturbed, planar shock.
On the starting vortex generated by a translating and rotating flat plate
- D. I. Pullin, John E. Sader
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- Journal:
- Journal of Fluid Mechanics / Volume 906 / 10 January 2021
- Published online by Cambridge University Press:
- 10 November 2020, A9
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We consider the trailing-edge vortex produced in an inviscid fluid by the start-up motion of a two-dimensional flat plate. A general starting motion is studied that includes the initial angle-of-attack of the plate (which may be zero), individual time power laws for plate translational and rotational speeds and the pivot position for plate rotation. A vortex-sheet representation for a start-up separated flow at the trailing edge is developed whose time-wise evolution is described by a Birkhoff–Rott equation coupled to an appropriate Kutta condition. This description includes convection by the outer flow, rotation and vortex-image self-induction. It admits a power-law similarity solution for the (small-time) primitive vortex, leading to an equation set where each term carries its own time-wise power-law factor. A set of four general plate motions is defined. Dominant-balance analysis of this set leads to discovery of three distinct start-up vortex-structure types that form the basis for all vortex motion. The properties of each type are developed in detail for some special cases. Numerical and analytical solutions are described and transition between solution types is discussed. Singular and degenerate vortex behaviour is discovered which may be due to the absence of fluid viscosity. An interesting case is start-up motion with zero initial angle of attack coupled to power-law plate rotation for which time-series examples are given that can be compared to high Reynolds number viscous flows.
The magnetised Richtmyer–Meshkov instability in two-fluid plasmas
- D. Bond, V. Wheatley, Y. Li, R. Samtaney, D. I. Pullin
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- Journal:
- Journal of Fluid Mechanics / Volume 903 / 25 November 2020
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- 30 September 2020, A41
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We investigate the effects of magnetisation on the two-fluid plasma Richtmyer–Meshkov instability of a single-mode thermal interface using a computational approach. The initial magnetic field is normal to the mean interface location. Results are presented for a magnetic interaction parameter of 0.1 and plasma skin depths ranging from 0.1 to 10 perturbation wavelengths. These are compared to initially unmagnetised and neutral fluid cases. The electron flow is found to be constrained to lie along the magnetic field lines resulting in significant longitudinal flow features that interact strongly with the ion fluid. The presence of an initial magnetic field is shown to suppress the growth of the initial interface perturbation with effectiveness determined by plasma length scale. Suppression of the instability is attributed to the magnetic field's contribution to the Lorentz force. This acts to rotate the vorticity vector in each fluid about the local magnetic-field vector leading to cyclic inversion and transport of the out-of-plane vorticity that drives perturbation growth. The transport of vorticity along field lines increases with decreasing plasma length scales and the wave packets responsible for vorticity transport begin to coalesce. In general, the two-fluid plasma Richtmyer–Meshkov instability is found to be suppressed through the action of the imposed magnetic field with increasing effectiveness as plasma length scale is decreased. For the conditions investigated, a critical skin depth for instability suppression is estimated.
Large-eddy simulation and modelling of Taylor–Couette flow
- W. Cheng, D. I. Pullin, R. Samtaney
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- Journal:
- Journal of Fluid Mechanics / Volume 890 / 10 May 2020
- Published online by Cambridge University Press:
- 12 March 2020, A17
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Wall-resolved large-eddy simulations (LES) of the incompressible Navier–Stokes equations together with empirical modelling for turbulent Taylor–Couette (TC) flow are presented. LES were performed with the inner cylinder rotating at angular velocity $\unicode[STIX]{x1D6FA}_{i}$ and the outer cylinder stationary. With $R_{i},R_{o}$ the inner and outer radii respectively, the radius ratio is $\unicode[STIX]{x1D702}=0.909$. The subgrid-scale stresses are represented using the stretched-vortex subgrid-scale model while the flow is resolved close to the wall. LES is implemented in the range $Re_{i}=10^{5}{-}10^{6}$ where $Re_{i}=\unicode[STIX]{x1D6FA}_{i}R_{i}d/\unicode[STIX]{x1D708}$ and $d=R_{o}-R_{i}$ is the cylinder gap. It is shown that the LES can capture the salient features of the flow, including the quantitative behaviour of spanwise Taylor rolls, the log variation in the inner-cylinder mean-velocity profile and the angular momentum redistribution due to the presence of Taylor rolls. A simple empirical model is developed for the turbulent, TC flow for both a stationary outer cylinder and also for co-rotating cylinders. This consists of near-wall, log-like turbulent wall layers separated by an annulus of constant angular momentum. Model results include the Nusselt number $Nu$ (torque required to maintain the flow) and measures of the wall-layer thickness as functions of both the Taylor number $Ta$ and $\unicode[STIX]{x1D702}$. These are compared with results from measurement, direct numerical simulation and the LES. A model extension to rough-wall turbulent flow is described. This shows an asymptotic, fully rough-wall state where the torque is independent of $Re_{i}/Ta$, and where $Nu\sim Ta^{1/2}$.
Large eddy simulation investigation of the canonical shock–turbulence interaction
- N. O. Braun, D. I. Pullin, D. I. Meiron
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- Journal of Fluid Mechanics / Volume 858 / 10 January 2019
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- 06 November 2018, pp. 500-535
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High resolution large eddy simulations (LES) are performed to study the interaction of a stationary shock with fully developed turbulent flow. Turbulent statistics downstream of the interaction are provided for a range of weakly compressible upstream turbulent Mach numbers $M_{t}=0.03{-}0.18$, shock Mach numbers $M_{s}=1.2{-}3.0$ and Taylor-based Reynolds numbers $Re_{\unicode[STIX]{x1D706}}=20{-}2500$. The LES displays minimal Reynolds number effects once an inertial range has developed for $Re_{\unicode[STIX]{x1D706}}>100$. The inertial range scales of the turbulence are shown to quickly return to isotropy, and downstream of sufficiently strong shocks this process generates a net transfer of energy from transverse into streamwise velocity fluctuations. The streamwise shock displacements are shown to approximately follow a $k^{-11/3}$ decay with wavenumber as predicted by linear analysis. In conjunction with other statistics this suggests that the instantaneous interaction of the shock with the upstream turbulence proceeds in an approximately linear manner, but nonlinear effects immediately downstream of the shock significantly modify the flow even at the lowest considered turbulent Mach numbers.
Large-eddy simulation of flow over a rotating cylinder: the lift crisis at $Re_{D}=6\times 10^{4}$
- W. Cheng, D. I. Pullin, R. Samtaney
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- Journal:
- Journal of Fluid Mechanics / Volume 855 / 25 November 2018
- Published online by Cambridge University Press:
- 19 September 2018, pp. 371-407
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We present wall-resolved large-eddy simulation (LES) of flow with free-stream velocity $\boldsymbol{U}_{\infty }$ over a cylinder of diameter $D$ rotating at constant angular velocity $\unicode[STIX]{x1D6FA}$, with the focus on the lift crisis, which takes place at relatively high Reynolds number $Re_{D}=U_{\infty }D/\unicode[STIX]{x1D708}$, where $\unicode[STIX]{x1D708}$ is the kinematic viscosity of the fluid. Two sets of LES are performed within the ($Re_{D}$, $\unicode[STIX]{x1D6FC}$)-plane with $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FA}D/(2U_{\infty })$ the dimensionless cylinder rotation speed. One set, at $Re_{D}=5000$, is used as a reference flow and does not exhibit a lift crisis. Our main LES varies $\unicode[STIX]{x1D6FC}$ in $0\leqslant \unicode[STIX]{x1D6FC}\leqslant 2.0$ at fixed $Re_{D}=6\times 10^{4}$. For $\unicode[STIX]{x1D6FC}$ in the range $\unicode[STIX]{x1D6FC}=0.48{-}0.6$ we find a lift crisis. This range is in agreement with experiment although the LES shows a deeper local minimum in the lift coefficient than the measured value. Diagnostics that include instantaneous surface portraits of the surface skin-friction vector field $\boldsymbol{C}_{\boldsymbol{f}}$, spanwise-averaged flow-streamline plots, and a statistical analysis of local, near-surface flow reversal show that, on the leeward-bottom cylinder surface, the flow experiences large-scale reorganization as $\unicode[STIX]{x1D6FC}$ increases through the lift crisis. At $\unicode[STIX]{x1D6FC}=0.48$ the primary-flow features comprise a shear layer separating from that side of the cylinder that moves with the free stream and a pattern of oscillatory but largely attached flow zones surrounded by scattered patches of local flow separation/reattachment on the lee and underside of the cylinder surface. Large-scale, unsteady vortex shedding is observed. At $\unicode[STIX]{x1D6FC}=0.6$ the flow has transitioned to a more ordered state where the small-scale separation/reattachment cells concentrate into a relatively narrow zone with largely attached flow elsewhere. This induces a low-pressure region which produces a sudden decrease in lift and hence the lift crisis. Through this process, the boundary layer does not show classical turbulence behaviour. As $\unicode[STIX]{x1D6FC}$ is further increased at constant $Re_{D}$, the localized separation zone dissipates with corresponding attached flow on most of the cylinder surface. The lift coefficient then resumes its increasing trend. A logarithmic region is found within the boundary layer at $\unicode[STIX]{x1D6FC}=1.0$.
Singularity formation on perturbed planar shock waves
- W. Mostert, D. I. Pullin, R. Samtaney, V. Wheatley
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- Journal:
- Journal of Fluid Mechanics / Volume 846 / 10 July 2018
- Published online by Cambridge University Press:
- 08 May 2018, pp. 536-562
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We present an analysis that predicts the time to development of a singularity in the shape profile of a spatially periodic perturbed, planar shock wave for ideal gas dynamics. Beginning with a formulation in complex coordinates of Whitham’s approximate model geometrical shock dynamics (GSD), we apply a spectral treatment to derive the asymptotic form for the leading-order behaviour of the shock Fourier coefficients for large mode numbers and time. This is shown to determine a critical time at which the coefficients begin to decay, with respect to mode number, at an algebraic rate with an exponent of $-5/2$ , indicating loss of analyticity and the formation of a singularity in the shock geometry. The critical time is found to be inversely proportional to a representative measure of perturbation amplitude $\unicode[STIX]{x1D716}$ with an explicit analytic form for the constant of proportionality in terms of gas and shock parameters. To leading order, the time to singularity formation is dependent only on the first Fourier mode. Comparison with results of numerical solutions to the full GSD equations shows that the predicted critical time somewhat underestimates the time for shock–shock (triple-point) formation, where the latter is obtained by post-processing the numerical GSD results using an edge-detection algorithm. Aspects of the analysis suggest that the appearance of loss of analyticity in the shock surface may be a precursor to the first appearance of shock–shocks, which may account for part of the discrepancy. The frequency of oscillation of the shock perturbation is accurately predicted. In addition, the analysis is extended to the evolution of a perturbed planar, fast magnetohydrodynamic shock for the case when the external magnetic field is aligned parallel to the unperturbed shock. It is found that, for a strong shock, the presence of the magnetic field produces only a higher-order correction to the GSD equations with the result that the time to loss of analyticity is the same as for the gas-dynamic flow. Limitations and improvements for the analysis are discussed, as are comparisons with the analogous appearance of singularity formation in vortex-sheet evolution in an incompressible inviscid fluid.
Large-eddy simulation of flow over a grooved cylinder up to transcritical Reynolds numbers
- W. Cheng, D. I. Pullin, R. Samtaney
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- Journal:
- Journal of Fluid Mechanics / Volume 835 / 25 January 2018
- Published online by Cambridge University Press:
- 27 November 2017, pp. 327-362
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We report wall-resolved large-eddy simulation (LES) of flow over a grooved cylinder up to the transcritical regime. The stretched-vortex subgrid-scale model is embedded in a general fourth-order finite-difference code discretization on a curvilinear mesh. In the present study $32$ grooves are equally distributed around the circumference of the cylinder, each of sinusoidal shape with height $\unicode[STIX]{x1D716}$, invariant in the spanwise direction. Based on the two parameters, $\unicode[STIX]{x1D716}/D$ and the Reynolds number $Re_{D}=U_{\infty }D/\unicode[STIX]{x1D708}$ where $U_{\infty }$ is the free-stream velocity, $D$ the diameter of the cylinder and $\unicode[STIX]{x1D708}$ the kinematic viscosity, two main sets of simulations are described. The first set varies $\unicode[STIX]{x1D716}/D$ from $0$ to $1/32$ while fixing $Re_{D}=3.9\times 10^{3}$. We study the flow deviation from the smooth-cylinder case, with emphasis on several important statistics such as the length of the mean-flow recirculation bubble $L_{B}$, the pressure coefficient $C_{p}$, the skin-friction coefficient $C_{f\unicode[STIX]{x1D703}}$ and the non-dimensional pressure gradient parameter $\unicode[STIX]{x1D6FD}$. It is found that, with increasing $\unicode[STIX]{x1D716}/D$ at fixed $Re_{D}$, some properties of the mean flow behave somewhat similarly to changes in the smooth-cylinder flow when $Re_{D}$ is increased. This includes shrinking $L_{B}$ and nearly constant minimum pressure coefficient. In contrast, while the non-dimensional pressure gradient parameter $\unicode[STIX]{x1D6FD}$ remains nearly constant for the front part of the smooth cylinder flow, $\unicode[STIX]{x1D6FD}$ shows an oscillatory variation for the grooved-cylinder case. The second main set of LES varies $Re_{D}$ from $3.9\times 10^{3}$ to $6\times 10^{4}$ with fixed $\unicode[STIX]{x1D716}/D=1/32$. It is found that this $Re_{D}$ range spans the subcritical and supercritical regimes and reaches the beginning of the transcritical flow regime. Mean-flow properties are diagnosed and compared with available experimental data including $C_{p}$ and the drag coefficient $C_{D}$. The timewise variation of the lift and drag coefficients are also studied to elucidate the transition among three regimes. Instantaneous images of the surface, skin-friction vector field and also of the three-dimensional Q-criterion field are utilized to further understand the dynamics of the near-surface flow structures and vortex shedding. Comparison of the grooved-cylinder flow with the equivalent flow over a smooth-wall cylinder shows structural similarities but significant differences. Both flows exhibit a clear common signature, which is the formation of mean-flow secondary separation bubbles that transform to other local flow features upstream of the main separation region (prior separation bubbles) as $Re_{D}$ is increased through the respective drag crises. Based on these similarities it is hypothesized that the drag crises known to occur for flow past a cylinder with different surface topographies is the result of a change in the global flow state generated by an interaction of primary flow separation with secondary flow recirculating motions that manifest as a mean-flow secondary bubble. For the smooth-wall flow this is accompanied by local boundary-layer flow transition to turbulence and a strong drag crisis, while for the grooved-cylinder case the flow remains laminar but unsteady through its drag crisis and into the early transcritical flow range.
Richtmyer–Meshkov instability of a thermal interface in a two-fluid plasma
- D. Bond, V. Wheatley, R. Samtaney, D. I. Pullin
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- Journal:
- Journal of Fluid Mechanics / Volume 833 / 25 December 2017
- Published online by Cambridge University Press:
- 03 November 2017, pp. 332-363
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We computationally investigate the Richtmyer–Meshkov instability of a density interface with a single-mode perturbation in a two-fluid, ion–electron plasma with no initial magnetic field. Self-generated magnetic fields arise subsequently. We study the case where the density jump across the initial interface is due to a thermal discontinuity, and select plasma parameters for which two-fluid plasma effects are expected to be significant in order to elucidate how they alter the instability. The instability is driven via a Riemann problem generated precursor electron shock that impacts the density interface ahead of the ion shock. The resultant charge separation and motion generates electromagnetic fields that cause the electron shock to degenerate and periodically accelerate the electron and ion interfaces, driving Rayleigh–Taylor instability. This generates small-scale structures and substantially increases interfacial growth over the hydrodynamic case.
Large-eddy simulation of flow over a cylinder with $Re_{D}$ from $3.9\times 10^{3}$ to $8.5\times 10^{5}$: a skin-friction perspective
- W. Cheng, D. I. Pullin, R. Samtaney, W. Zhang, W. Gao
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- Journal:
- Journal of Fluid Mechanics / Volume 820 / 10 June 2017
- Published online by Cambridge University Press:
- 05 May 2017, pp. 121-158
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We present wall-resolved large-eddy simulations (LES) of flow over a smooth-wall circular cylinder up to $Re_{D}=8.5\times 10^{5}$, where $Re_{D}$ is Reynolds number based on the cylinder diameter $D$ and the free-stream speed $U_{\infty }$. The stretched-vortex subgrid-scale (SGS) model is used in the entire simulation domain. For the sub-critical regime, six cases are implemented with $3.9\times 10^{3}\leqslant Re_{D}\leqslant 10^{5}$. Results are compared with experimental data for both the wall-pressure-coefficient distribution on the cylinder surface, which dominates the drag coefficient, and the skin-friction coefficient, which clearly correlates with the separation behaviour. In the super-critical regime, LES for three values of $Re_{D}$ are carried out at different resolutions. The drag-crisis phenomenon is well captured. For lower resolution, numerical discretization fluctuations are sufficient to stimulate transition, while for higher resolution, an applied boundary-layer perturbation is found to be necessary to stimulate transition. Large-eddy simulation results at $Re_{D}=8.5\times 10^{5}$, with a mesh of $8192\times 1024\times 256$, agree well with the classic experimental measurements of Achenbach (J. Fluid Mech., vol. 34, 1968, pp. 625–639) especially for the skin-friction coefficient, where a spike is produced by the laminar–turbulent transition on the top of a prior separation bubble. We document the properties of the attached-flow boundary layer on the cylinder surface as these vary with $Re_{D}$. Within the separated portion of the flow, mean-flow separation–reattachment bubbles are observed at some values of $Re_{D}$, with separation characteristics that are consistent with experimental observations. Time sequences of instantaneous surface portraits of vector skin-friction trajectory fields indicate that the unsteady counterpart of a mean-flow separation–reattachment bubble corresponds to the formation of local flow-reattachment cells, visible as coherent bundles of diverging surface streamlines.
Rough-wall turbulent boundary layers with constant skin friction
- A. Sridhar, D. I. Pullin, W. Cheng
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- Journal:
- Journal of Fluid Mechanics / Volume 818 / 10 May 2017
- Published online by Cambridge University Press:
- 28 March 2017, pp. 26-45
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A semi-empirical model is presented that describes the development of a fully developed turbulent boundary layer in the presence of surface roughness with length scale $k_{s}$ that varies with streamwise distance $x$. Interest is centred on flows for which all terms of the von Kármán integral relation, including the ratio of outer velocity to friction velocity $U_{\infty }^{+}\equiv U_{\infty }/u_{\unicode[STIX]{x1D70F}}$, are streamwise constant. For $Re_{x}$ assumed large, use is made of a simple log-wake model of the local turbulent mean-velocity profile that contains a standard mean-velocity correction for the asymptotic fully rough regime and with assumed constant parameter values. It is then shown that, for a general power-law external velocity variation $U_{\infty }\sim x^{m}$, all measures of the boundary-layer thickness must be proportional to $x$ and that the surface sand-grain roughness scale variation must be the linear form $k_{s}(x)=\unicode[STIX]{x1D6FC}x$, where $x$ is the distance from the boundary layer of zero thickness and $\unicode[STIX]{x1D6FC}$ is a dimensionless constant. This is shown to give a two-parameter $(m,\unicode[STIX]{x1D6FC})$ family of solutions, for which $U_{\infty }^{+}$ (or equivalently $C_{f}$) and boundary-layer thicknesses can be simply calculated. These correspond to perfectly self-similar boundary-layer growth in the streamwise direction with similarity variable $z/(\unicode[STIX]{x1D6FC}x)$, where $z$ is the wall-normal coordinate. Results from this model over a range of $\unicode[STIX]{x1D6FC}$ are discussed for several cases, including the zero-pressure-gradient ($m=0$) and sink-flow ($m=-1$) boundary layers. Trends observed in the model are supported by wall-modelled large-eddy simulation of the zero-pressure-gradient case for $Re_{x}$ in the range $10^{8}{-}10^{10}$ and for four values of $\unicode[STIX]{x1D6FC}$. Linear streamwise growth of the displacement, momentum and nominal boundary-layer thicknesses is confirmed, while, for each $\unicode[STIX]{x1D6FC}$, the mean-velocity profiles and streamwise turbulent variances are found to collapse reasonably well onto $z/(\unicode[STIX]{x1D6FC}x)$. For given $\unicode[STIX]{x1D6FC}$, calculations of $U_{\infty }^{+}$ obtained from large-eddy simulations are streamwise constant and independent of $Re_{x}$ when this is large. The present results suggest that, in the sense that $U_{\infty }^{+}(\unicode[STIX]{x1D6FC},m)$ is constant, these flows can be interpreted as the fully rough limit for boundary layers in the presence of small-scale linear roughness.
Geometrical shock dynamics for magnetohydrodynamic fast shocks
- W. Mostert, D. I. Pullin, R. Samtaney, V. Wheatley
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- Journal:
- Journal of Fluid Mechanics / Volume 811 / 25 January 2017
- Published online by Cambridge University Press:
- 12 December 2016, R2
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We describe a formulation of two-dimensional geometrical shock dynamics (GSD) suitable for ideal magnetohydrodynamic (MHD) fast shocks under magnetic fields of general strength and orientation. The resulting area–Mach-number–shock-angle relation is then incorporated into a numerical method using pseudospectral differentiation. The MHD-GSD model is verified by comparison with results from nonlinear finite-volume solution of the complete ideal MHD equations applied to a shock implosion flow in the presence of an oblique and spatially varying magnetic field ahead of the shock. Results from application of the MHD-GSD equations to the stability of fast MHD shocks in two dimensions are presented. It is shown that the time to formation of triple points for both perturbed MHD and gas-dynamic shocks increases as $\unicode[STIX]{x1D716}^{-1}$, where $\unicode[STIX]{x1D716}$ is a measure of the initial Mach-number perturbation. Symmetry breaking in the MHD case is demonstrated. In cylindrical converging geometry, in the presence of an azimuthal field produced by a line current, the MHD shock behaves in the mean as in Pullin et al. (Phys. Fluids, vol. 26, 2014, 097103), but suffers a greater relative pressure fluctuation along the shock than the gas-dynamic shock.
Converging cylindrical magnetohydrodynamic shock collapse onto a power-law-varying line current
- W. Mostert, D. I. Pullin, R. Samtaney, V. Wheatley
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- Journal:
- Journal of Fluid Mechanics / Volume 793 / 25 April 2016
- Published online by Cambridge University Press:
- 16 March 2016, pp. 414-443
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We investigate the convergence behaviour of a cylindrical, fast magnetohydrodynamic (MHD) shock wave in a neutrally ionized gas collapsing onto an axial line current that generates a power law in time, azimuthal magnetic field. The analysis is done within the framework of a modified version of ideal MHD for an inviscid, non-dissipative, neutrally ionized compressible gas. The time variation of the magnetic field is tuned such that it approaches zero at the instant that the shock reaches the axis. This configuration is motivated by the desire to produce a finite magnetic field at finite shock radius but a singular gas pressure and temperature at the instant of shock impact. Our main focus is on the variation with shock radius $r$, as $r\rightarrow 0$, of the shock Mach number $M(r)$ and pressure behind the shock $p(r)$ as a function of the magnetic field power-law exponent ${\it\mu}\geqslant 0$, where ${\it\mu}=0$ gives a constant-in-time line current. The flow problem is first formulated using an extension of geometrical shock dynamics (GSD) into the time domain to take account of the time-varying conditions ahead of the converging shock, coupled with appropriate shock-jump conditions for a fast, symmetric MHD shock. This provides a pair of ordinary differential equations describing both $M(r)$ and the time evolution on the shock, as a function of $r$, constrained by a collapse condition required to achieve tuned shock convergence. Asymptotic, analytical results for $M(r)$ and $p(r)$ are obtained over a range of ${\it\mu}$ for general ${\it\gamma}$, and for both small and large $r$. In addition, numerical solutions of the GSD equations are performed over a large range of $r$, for selected parameters using ${\it\gamma}=5/3$. The accuracy of the GSD model is verified for some cases using direct numerical solution of the full, radially symmetric MHD equations using a shock-capturing method. For the GSD solutions, it is found that the physical character of the shock convergence to the axis is a strong function of ${\it\mu}$. For $0\leqslant {\it\mu}<4/13$, $M$ and $p$ both approach unity at shock impact $r=0$ owing to the dominance of the strong magnetic field over the amplifying effects of geometrical convergence. When ${\it\mu}\geqslant 0.816$ (for ${\it\gamma}=5/3$), geometrical convergence is dominant and the shock behaves similarly to a converging gas dynamic shock with singular $M(r)$ and $p(r)$, $r\rightarrow 0$. For $4/13<{\it\mu}\leqslant 0.816$ three distinct regions of $M(r)$ variation are identified. For each of these $p(r)$ is singular at the axis.
Large-eddy simulation of separation and reattachment of a flat plate turbulent boundary layer
- W. Cheng, D. I. Pullin, R. Samtaney
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- Journal:
- Journal of Fluid Mechanics / Volume 785 / 25 December 2015
- Published online by Cambridge University Press:
- 11 November 2015, pp. 78-108
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We present large-eddy simulations (LES) of separation and reattachment of a flat-plate turbulent boundary-layer flow. Instead of resolving the near wall region, we develop a two-dimensional virtual wall model which can calculate the time- and space-dependent skin-friction vector field at the wall, at the resolved scale. By combining the virtual-wall model with the stretched-vortex subgrid-scale (SGS) model, we construct a self-consistent framework for the LES of separating and reattaching turbulent wall-bounded flows at large Reynolds numbers. The present LES methodology is applied to two different experimental flows designed to produce separation/reattachment of a flat-plate turbulent boundary layer at medium Reynolds number $Re_{{\it\theta}}$ based on the momentum boundary-layer thickness ${\it\theta}$. Comparison with data from the first case at $Re_{{\it\theta}}=2000$ demonstrates the present capability for accurate calculation of the variation, with the streamwise co-ordinate up to separation, of the skin friction coefficient, $Re_{{\it\theta}}$, the boundary-layer shape factor and a non-dimensional pressure-gradient parameter. Additionally the main large-scale features of the separation bubble, including the mean streamwise velocity profiles, show good agreement with experiment. At the larger $Re_{{\it\theta}}=11\,000$ of the second case, the LES provides good postdiction of the measured skin-friction variation along the whole streamwise extent of the experiment, consisting of a very strong adverse pressure gradient leading to separation within the separation bubble itself, and in the recovering or reattachment region of strongly-favourable pressure gradient. Overall, the present two-dimensional wall model used in LES appears to be capable of capturing the quantitative features of a separation-reattachment turbulent boundary-layer flow at low to moderately large Reynolds numbers.
Turbulent mixing driven by spherical implosions. Part 1. Flow description and mixing-layer growth
- M. Lombardini, D. I. Pullin, D. I. Meiron
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- Journal:
- Journal of Fluid Mechanics / Volume 748 / 10 June 2014
- Published online by Cambridge University Press:
- 28 April 2014, pp. 85-112
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We present large-eddy simulations (LES) of turbulent mixing at a perturbed, spherical interface separating two fluids of differing densities and subsequently impacted by a spherically imploding shock wave. This paper focuses on the differences between two fundamental configurations, keeping fixed the initial shock Mach number ${\approx }1.2$, the density ratio (precisely $|A_0|\approx 0.67$) and the perturbation shape (dominant spherical wavenumber $\ell _0=40$ and amplitude-to-initial radius of $3\, \%$): the incident shock travels from the lighter fluid to the heavy fluid or, inversely, from the heavy to the light fluid. After describing the computational problem we present results on the radially symmetric flow, the mean flow, and the growth of the mixing layer. Turbulent statistics are developed in Part 2 (Lombardini, M., Pullin, D. I. & Meiron, D. I. J. Fluid Mech., vol. 748, 2014, pp. 113–142). A wave-diagram analysis of the radially symmetric flow highlights that the light–heavy mixing layer is processed by consecutive reshocks, and not by reverberating rarefaction waves as is usually observed in planar geometry. Less surprisingly, reshocks process the heavy–light mixing layer as in the planar case. In both configurations, the incident imploding shock and the reshocks induce Richtmyer–Meshkov (RM) instabilities at the density layer. However, we observe differences in the mixing-layer growth because the RM instability occurrences, Rayleigh–Taylor (RT) unstable scenarios (due to the radially accelerated motion of the layer) and phase inversion events are different. A small-amplitude stability analysis along the lines of Bell (Los Alamos Scientific Laboratory Report, LA-1321, 1951) and Plesset (J. Appl. Phys., vol. 25, 1954, pp. 96–98) helps quantify the effects of the mean flow on the mixing-layer growth by decoupling the effects of RT/RM instabilities from Bell–Plesset effects associated with geometric convergence and compressibility for arbitrary convergence ratios. The analysis indicates that baroclinic instabilities are the dominant effect, considering the low convergence ratio (${\approx } 2$) and rather high ($\ell >10$) mode numbers considered.
Turbulent mixing driven by spherical implosions. Part 2. Turbulence statistics
- M. Lombardini, D. I. Pullin, D. I. Meiron
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- Journal of Fluid Mechanics / Volume 748 / 10 June 2014
- Published online by Cambridge University Press:
- 28 April 2014, pp. 113-142
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We present large-eddy simulations (LES) of turbulent mixing at a perturbed, spherical interface separating two fluids of differing densities and subsequently impacted by a spherically imploding shock wave. This paper focuses on the differences between two fundamental configurations, keeping fixed the initial shock Mach number $\approx $1.2, the density ratio (precisely $|A_0|\approx 0.67$) and the perturbation shape (dominant spherical wavenumber $\ell _0=40$ and amplitude-to-initial radius of 3 %): the incident shock travels from the lighter fluid to the heavy one, or inversely, from the heavy to the light fluid. In Part 1 (Lombardini, M., Pullin, D. I. & Meiron, D. I., J. Fluid Mech., vol. 748, 2014, pp. 85–112), we described the computational problem and presented results on the radially symmetric flow, the mean flow, and the growth of the mixing layer. In particular, it was shown that both configurations reach similar convergence ratios $\approx $2. Here, turbulent mixing is studied through various turbulence statistics. The mixing activity is first measured through two mixing parameters, the mixing fraction parameter $\varTheta $ and the effective Atwood ratio $A_e$, which reach similar late time values in both light–heavy and heavy–light configurations. The Taylor-scale Reynolds numbers attained at late times are estimated at approximately 2000 in the light–heavy case and 1000 in the heavy–light case. An analysis of the density self-correlation $b$, a fundamental quantity in the study of variable-density turbulence, shows asymmetries in the mixing layer and non-Boussinesq effects generally observed in high-Reynolds-number Rayleigh–Taylor (RT) turbulence. These traits are more pronounced in the light–heavy mixing layer, as a result of its flow history, in particular because of RT-unstable phases (see Part 1). Another measure distinguishing light–heavy from heavy–light mixing is the velocity-to-scalar Taylor microscales ratio. In particular, at late times, larger values of this ratio are reported in the heavy–light case. The late-time mixing displays the traits some of the traits of the decaying turbulence observed in planar Richtmyer–Meshkov (RM) flows. Only partial isotropization of the flow (in the sense of turbulent kinetic energy (TKE) and dissipation) is observed at late times, the Reynolds normal stresses (and, thus, the directional Taylor microscales) being anisotropic while the directional Kolmogorov microscales approach isotropy. A spectral analysis is developed for the general study of statistically isotropic turbulent fields on a spherical surface, and applied to the present flow. The resulting angular power spectra show the development of an inertial subrange approaching a Kolmogorov-like $-5/3$ power law at high wavenumbers, similarly to the scaling obtained in planar geometry. It confirms the findings of Thomas & Kares (Phys. Rev. Lett., vol. 109, 2012, 075004) at higher convergence ratios and indicates that the turbulent scales do not seem to feel the effect of the spherical mixing-layer curvature.
Transition to turbulence in shock-driven mixing: a Mach number study
- M. Lombardini, D. I. Pullin, D. I. Meiron
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- Journal of Fluid Mechanics / Volume 690 / 10 January 2012
- Published online by Cambridge University Press:
- 21 November 2011, pp. 203-226
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Large-eddy simulations of single-shock-driven mixing suggest that, for sufficiently high incident Mach numbers, a two-gas mixing layer ultimately evolves to a late-time, fully developed turbulent flow, with Kolmogorov-like inertial subrange following a power law. After estimating the kinetic energy injected into the diffuse density layer during the initial shock–interface interaction, we propose a semi-empirical characterization of fully developed turbulence in such flows, based on scale separation, as a function of the initial parameter space, as , which corresponds to late-time Taylor-scale Reynolds numbers . In this expression, represents the post-shock perturbation amplitude, the change in interface velocity induced by the shock refraction, the characteristic kinematic viscosity of the mixture, the inner diffuse thickness of the initial density profile, the post-shock Atwood ratio, and for the gas combination and post-shock perturbation amplitude considered. The initially perturbed interface separating air and SF6 (pre-shock Atwood ratio ) was impacted in a heavy–light configuration by a shock wave of Mach number , 1.25, 1.56, 3.0 or 5.0, for which is fixed at about 25 % of the dominant wavelength of an initial, Gaussian perturbation spectrum. Only partial isotropization of the flow (in the sense of turbulent kinetic energy and dissipation) is observed during the late-time evolution of the mixing zone. For all Mach numbers considered, the late-time flow resembles homogeneous decaying turbulence of Batchelor type, with a turbulent kinetic energy decay exponent and large-scale () energy spectrum , and a molecular mixing fraction parameter, . An appropriate time scale characterizing the Taylor-scale Reynolds number decay, as well as the evolution of mixing parameters such as and the effective Atwood ratio , seem to indicate the existence of low- and high-Mach-number regimes.
Evolution of vortex-surface fields in viscous Taylor–Green and Kida–Pelz flows
- Yue Yang, D. I. Pullin
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- Journal of Fluid Mechanics / Volume 685 / 25 October 2011
- Published online by Cambridge University Press:
- 06 October 2011, pp. 146-164
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In order to investigate continuous vortex dynamics based on a Lagrangian-like formulation, we develop a theoretical framework and a numerical method for computation of the evolution of a vortex-surface field (VSF) in viscous incompressible flows with simple topology and geometry. Equations describing the continuous, timewise evolution of a VSF from an existing VSF at an initial time are first reviewed. Non-uniqueness in this formulation is resolved by the introduction of a pseudo-time and a corresponding pseudo-evolution in which the evolved field is ‘advected’ by frozen vorticity onto a VSF. A weighted essentially non-oscillatory (WENO) method is used to solve the pseudo-evolution equations in pseudo-time, providing a dissipative-like regularization. Vortex surfaces are then extracted as iso-surfaces of the VSFs at different real physical times. The method is applied to two viscous flows with Taylor–Green and Kida–Pelz initial conditions respectively. Results show the collapse of vortex surfaces, vortex reconnection, the formation and roll-up of vortex tubes, vorticity intensification between anti-parallel vortex tubes, and vortex stretching and twisting. A possible scenario for understanding the transition from a smooth laminar flow to turbulent flow in terms of topology of vortex surfaces is discussed.
Large-eddy simulation of the zero-pressure-gradient turbulent boundary layer up to Reθ = O(1012)
- M. Inoue, D. I. Pullin
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- Journal:
- Journal of Fluid Mechanics / Volume 686 / 10 November 2011
- Published online by Cambridge University Press:
- 29 September 2011, pp. 507-533
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A near-wall subgrid-scale (SGS) model is used to perform large-eddy simulation (LES) of the developing, smooth-wall, zero-pressure-gradient flat-plate turbulent boundary layer. In this model, the stretched-vortex, SGS closure is utilized in conjunction with a tailored, near-wall model designed to incorporate anisotropic vorticity scales in the presence of the wall. Large-eddy simulations of the turbulent boundary layer are reported at Reynolds numbers based on the free-stream velocity and the momentum thickness in the range . Results include the inverse square-root skin-friction coefficient, , velocity profiles, the shape factor , the von Kármán ‘constant’ and the Coles wake factor as functions of . Comparisons with some direct numerical simulation (DNS) and experiment are made including turbulent intensity data from atmospheric-layer measurements at . At extremely large , the empirical Coles–Fernholz relation for skin-friction coefficient provides a reasonable representation of the LES predictions. While the present LES methodology cannot probe the structure of the near-wall region, the present results show turbulence intensities that scale on the wall-friction velocity and on the Clauser length scale over almost all of the outer boundary layer. It is argued that LES is suggestive of the asymptotic, infinite Reynolds number limit for the smooth-wall turbulent boundary layer and different ways in which this limit can be approached are discussed. The maximum of the present simulations appears to be limited by machine precision and it is speculated, but not demonstrated, that even larger could be achieved with quad- or higher-precision arithmetic.